What is the formal definition of 'opposite sides' of a polygon? Does a
regular pentagon have opposite sides? Does a concave polygon have
opposite sides? How can we define it consistent with our intuition?

It seems that sometimes we use the not-equal sign to state an absolute
inequality (two expressions always have a different value), and
sometimes we just mean to say that two expressions are not always
equal. Can you please clarify this apparent ambiguity? How do we
know which meaning is being used?

I hope to become a chef someday, and I would like to know how algebra
(specifically polynomials, the quadratic formula, factoring, etc.) could
possibly relate to my future career and my life beyond high school.

Why do so many of today's mathematicians present their work in such a
formal way, hiding their ideas behind all sorts of notation,
definitions and the like? Why is there so little explanation of the
ideas and concepts? I like math, but I find this aspect of it very
annoying.

Is there any difference between a science project and a math project?
We have a science fair coming up and I wanted to do some math, but my
friend says that the scientific process is very different from math.